Mechanisms of light harvesting by photosystem II inplants

Kapil Amarnath,1∗ Doran I. G. Bennett,1∗ Anna R. Schneider,2

Graham R. Fleming1∗

1Department of Chemistry, University of California, Berkeley, CAand Physical Biosciences Division, Lawrence Berkeley National Lab, Berkeley, CA

2Biophysics Graduate Group, University of California, Berkeley

∗To whom correspondence should be addressed; E-mail: kapil@alum.mit.edu,bennett.doran@gmail.com, grfleming@lbl.gov

Light harvesting by photosystem II (PSII) in plants is highly efficient and accli-

mates to rapid changes in the intensity of sunlight. However, the mechanisms

of PSII light harvesting have remained experimentally inaccessible. Using a

structure-based model of excitation energy flow in 200 nanometer (nm) x 200

nm patches of the grana membrane, where PSII is located, we accurately simu-

lated chlorophyll fluorescence decay data with no free parameters. Excitation

movement through the light harvesting antenna is diffusive, but becomes subd-

iffusive in the presence of charge separation at reaction centers. The influence

of membrane morphology on light harvesting efficiency is determined by the

excitation diffusion length of 50 nm in the antenna. Our model provides the

basis for understanding how nonphotochemical quenching mechanisms affect

PSII light harvesting in grana membranes.

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Main Text

Photosynthetic light harvesting is a multi-scale process that spans from tens of femtoseconds

to minutes, and from angstroms to hundreds of nanometers (1). In green plants, photosynthetic

light harvesting is performed by photosystems I and II, which are located in separate regions

of the thylakoid membrane (2). Photosystem II (PSII), which provides the electrons that drive

photosynthesis, is located in the grana stacks of the thylakoid. PSII light harvesting starts when

pigments bound to antenna proteins absorb sunlight. The nascent excitation energy is trans-

ferred to reaction centers (RC) where it is converted to chemical energy via charge separation.

PSII is both an efficient and adaptable light harvesting material. At maximum light harvesting

capacity, the average time for excitation capture by PSII (∼300 ps (3, 4)) is significantly less

than the excited state lifetime of pigments in the grana (∼2 ns (5, 6)), which results in a >80%

efficiency for converting absorbed sunlight into chemical energy (7). In addition, PSII accli-

mates to changes in sunlight intensity (8), wavelength (9), and downstream metabolism (10),

which occur on the millisecond to minutes timescale. The final outcome of excitation in the

grana is determined by the interplay between the rates of excitation energy transfer through

the antenna and processes that irreversibly quench excitation (e.g. fluorescence, non-radiative

processes, and productive photochemistry in the RC). Time-resolved chlorophyll fluorescence,

which is currently the primary technique for measuring PSII light harvesting in a variety of en-

vironmental conditions in vivo (11), does not have sufficient spatiotemporal resolution to fully

characterize the light harvesting kinetics of PSII (12). Thus a model for PSII light harvesting

that accurately captures its underlying dynamics is needed to understand the physical principles

that give rise to its functionality. Such principles are essential for understanding the molecular

mechanisms of photosynthesis and could be used to engineer artificial light harvesting systems

with properties that mimic PSII (13–15).

2

Current models of light harvesting in PSII treat the rates of energy transfer and trapping as

parameters whose values are determined by fitting to chlorophyll fluorescence data (4, 16, 17).

Conceptually, these models mainly fit within the canonical “lake” and “puddle” models (18),

which are currently used to describe the dynamics of energy transfer and trapping by PSII on the

hundreds of picoseconds time scale in grana membranes. In the lake model, each excitation tra-

verses the grana membrane sufficiently quickly to access all RCs equally. In the puddle model,

excitation movement is spatially limited to the smallest photosynthetic unit, which is thought

to be a photosystem II supercomplex (PSII-S) and a few surrounding major light harvesting

complex II (LHCII) antenna (Fig. 1A, inset) (19). PSII-S consists of two reaction centers and

a small number of antenna proteins (20). Up to now, models with free parameters for energy

transfer and trapping in PSII have been considered accurate if they can fit chlorophyll fluo-

rescence decay data well. However, multiple models that implicitly assume either the lake or

puddle model can fit chlorophyll fluorescence decay data equally well (4, 16). As a result, it

remains unclear whether the fitted rates found by these methods are physically meaningful and,

more broadly, to what extent either the lake or puddle models are appropriate for describing

PSII light harvesting. It is well-established that simulating the ultrafast excitation energy trans-

fer dynamics in individual pigment-protein complexes requires a structure-based approach that

correctly accounts for each chromophore (21, 22). We previously showed that this approach is

required even in a small group of pigment-protein complexes in our model of light harvesting

within isolated PSII-S (23). Here, we demonstrate that an accurate model for PSII light har-

vesting in grana membranes must be based on a correct description of the picosecond dynamics

that occur at the nanometer scale of individual pigments.

We constructed a parameter-less model for PSII light harvesting that is firmly based on an

accurate physical description of the nanoscopic energy transfer dynamics in grana membranes.

We performed Monte Carlo simulations on 200 nm x 200 nm patches of the grana membrane

3

(24) (see Methods, below) to generate examples of the mixed (Fig. 1A) and segregated (Fig. 1B)

organizations previously observed (2, 26). In the segregated membrane, PSII-S and LHCII

separate into so-called PSII-S crystalline arrays and LHCII aggregates. The Monte Carlo model

contains a small number of energetic interactions based on in vivo phenomenology and provides

plausible locations of LHCII in grana, which are difficult to visualize using existing imaging

techniques (24). We superimposed the crystal structures of the chlorophyll pigments in PSII-

S (20) and LHCII (27) on these simulations to establish the locations of all of the chlorophylls in

a grana patch (Fig. 1A, inset). In previous work on PSII-S, we grouped chlorophylls into highly

energetically coupled clusters called domains and demonstrated that it is sufficient to describe

the excitation dynamics at the domain level (23). We assume that the kinetics within LHCII

and PSII-S are the same as calculated previously on isolated complexes (23). Inhomogeneously

averaged rates of energy transfer between domains on different complexes were calculated using

Generalized Forster theory (Methods). We modeled photochemistry in the reaction centers with

two phenomenological “radical pair” (RP) states. The rate constants between the RP states are

the same as those established in previous work on the PSII-S (23). Using our method we can

routinely construct a rate matrix for grana membrane patches that contain more pigments (up to

105) than any natural light harvesting system previously modeled.

Our model accurately simulates chlorophyll fluorescence data and demonstrates that ex-

tracting the amplitude and lifetime components from a simple fit does not capture the complex

kinetics of PSII light harvesting. The comparison of the simulation of the mixed membrane

(black solid line) with experimental data from thylakoids in ref. (3) (red dotted line) is shown in

Fig. 1C. We expected the mixed membrane to be the dominant morphology of the measured thy-

lakoids based on the conditions in which the plants (Arabidopsis thaliana) were grown (3, 28).

The excellent agreement of the simulation with the experimental data, which was not guaranteed

a priori, suggests that our parameter-free model correctly describes excitation energy transport

4

during PSII light harvesting. The simulated decay can be fit well to a sum of a few exponentials

(Fig. 1C, inset, green bars), as is frequently done to extract the amplitude and lifetime compo-

nents (11). However, the fit does not capture the complex distribution calculated from the rate

matrix (Fig. 1C, inset, black bars). Fit-based models that are only sufficiently complex to fit

chlorophyll fluorescence decays well will not accurately characterize the underlying PSII light

harvesting dynamics.

Our simulation of the spatiotemporal dynamics of chlorophyll excitation in the grana al-

lowed us to examine the assumptions underpinning the lake and puddle models. Upon uniform

initial excitation of either the mixed or segregated membranes, the lake model predicts that the

excitation distribution will quickly reach a steady-state spatial distribution. However, we did

not observe a steady-state distribution for either the mixed (Movie S1) or segregated (Movie

S2) membranes. The puddle model predicts that there are a few (2-3) different types of trapping

dynamics at reaction centers throughout the membrane. On the contrary, we observed trapping

dynamics (Movies S3 and S4) that vary considerably between different PSII-S. Neither the lake

nor the puddle models accurately describe excitation dynamics on the hundreds of picoseconds

timescale and hundreds of nanometers length scale of the grana membrane.

To understand how excitation moves through the grana, we simulated excitation energy flow

from single pigment-protein complexes in LHCII aggregates, PSII crystalline arrays, and mixed

membranes (Movies S5-S9). To determine the effect of charge separation on excitation move-

ment, we simulated the PSII crystalline arrays and mixed membranes both with and without

the radical pair (RP) states in the reaction centers. We quantified excitation transport in these

simulations by calculating the time-dependence of the variance of the excitation probability

distribution (Fig. 2A). Transport across the grana was well described by fitting the equation

σ2(t)− σ2(0) = Atα, (1)

5

Figure 1: Structure-based modeling of energy transfer is required to accurately simu-late PSII light harvesting. (A) and (B) The representative mixed and segregated membranemorphologies generated using Monte Carlo simulations and used throughout this work. PSIIsupercomplexes (PSII-S) are indicated by the teal discorectangles, while major light harvestingantenna complexes (LHCII) are indicated by the green circles. The segregated membrane formsPSII-S crystalline arrays and LHCII aggregates. As shown schematically in the inset in (A), ex-isting crystal structures of PSII-S (20) and LHCII (27) were overlaid on these membrane patchesto establish the locations of all chlorophyll pigments. The teal and green dashed lines outline theexcluded area associated with PSII-S and LHCII in the Monte Carlo simulations, respectively.The chlorophyll pigments are indicated in blue, while the protein is depicted by the cartoonribbon. PSII-S is a 2-fold symmetric dimer of pigment-protein complexes that are outlined byblack lines. LHCII-s, CP26, CP29, CP43, and CP47 are antenna proteins, while RC indicatesthe reaction center. The inhomogeneously-averaged rates of energy transfer between strongly-coupled clusters of pigments were calculated using Generalized Forster theory. (C) Simulatedfluorescence decay of the mixed membrane (solid black line) and the PSII-component of exper-imental fluorescence decay data from thylakoid membranes from ref. (3) (red, dotted line). Theinset shows the lifetime components and amplitudes of the simulated decay as calculated usingour model (black bars) or by fitting to three exponential decays (green bars).

6

where σ2(t) is the variance at time t, σ2(0) is the variance of the initial distribution of excitation,

and A and α are fit parameters (Fig. S4B). For α = 1, transport is diffusive, while subdiffusive

transport results if α is significantly less than 1. For the LHCII aggregate (Fig. 2A, green tri-

angles) and for the mixed membranes and PSII crystalline arrays without RP states (Fig. 2A,

red and blue dashed lines, respectively), α ≈ 1 and thus transport within the antenna can be

considered diffusive. However, transport for both the mixed and PSII crystal cases with RP

states (Fig. 2A, red and blue solid lines, respectively) is sub-diffusive (α < 0.75). Subdiffu-

sivity can occur when the energetic differences between sites is on the order of or greater than

kBT . We calculated the ∆G for the RC→RP1 (radical pair 1 ) step to be -5.5kBT on the basis

of our previously published rates (23). Thus, RP1 serves as an energetic trap that causes sub-

diffusive transport. The energy transfer rates in our model are averaged over inhomogeneous

realizations, which could mitigate the slow-down of diffusion that occurs when the width of the

inhomogeneous distribution is greater than kBT (29). The largest standard deviation of exciton

energies across an inhomogeneous distribution in our model is 107 cm−1, which is significantly

less than the value of kBT at room temperature (210 cm−1). We note that the diffusive picture of

excitation energy transport elaborated here is consistent with predictions made from the PSII-S

model (23) and suggests that, in the grana, excitation energy flows neither “directionally” nor

energetically downhill on “preferred pathways” (30).

We calculated the excitation diffusion length, LD, and the diffusivity, D, to reduce the

dynamics observed above to single parameters that can be used to qualitatively understand light

harvesting behavior in grana.

σ2(t)− σ2(0) = 4Dt = L2 (2)

A plot of D as a function of time is shown in Fig. S4A. In the case of subdiffusive transport,

the diffusivities decrease in time as excitations are held by the RP1 trap state. The diffusion

7

Figure 2: Excitation transport in grana membranes. Simulation of excitation movementin the five grana membrane configurations shown in the legend: mixed membrane with andwithout the radical pair (RP) states in the reaction center, PSII-S crystalline array (PSII-S c.a.)with and without the RP states, and LHCII aggregate (LHCII agg). In each case, excitationwas initiated on a single pigment-protein complex. (A) The change in the spread of excitationover time. The diffusion exponent α (see text for details) is shown on the right of the plot.(B) Fraction of surviving excitation as a function of net displacement L (eq. 2) from the initialstarting point. The dashed line, where the fraction of surviving excitation is 1/e, demarcates theexcitation diffusion length (LD). The dimensions of some of the configurations were too smallto calculate an LD, so linear extrapolation was used to approximate it (line segments that do notinclude markers).

8

constants for the three cases ranged from 1−5×10−3 cm2/sec. These values agree with previous

experimental work, which used singlet-singlet annihilation measurements of chloroplasts to

suggest a lower bound of D ≈ 10−3 cm−2/sec (31). The fraction of excitation remaining as a

function of the net spatial displacement, L, is shown in Fig. 2B. LD is defined, by convention, as

the minimum net displacement in one dimension achieved by 37% of the excitation population.

The LD for the mixed membrane and the PSII crystal with radical pair states were ∼25 nm

and ∼15 nm, respectively, though there was some variation depending on the starting point of

excitation (Fig. S5). The LD for the LHCII aggregate, mixed membrane, and PSII crystalline

array without radical pair states was ∼50 nm. The values of LD and D in the antenna compare

favorably with the values observed thus far in organic thin films (32) and quantum dot arrays

(29).

Using the excitation diffusion length (LD), we explored the longstanding question of how

membrane morphology influences the efficiency with which PSII converts absorbed light into

chemical energy, the photochemical yield (2, 30). We calculated the photochemical yield in

both the mixed and segregated cases to be 0.82 and 0.70, respectively, upon spatially uniform

excitation across the membranes. The calculated value of 0.82 for the mixed membrane is

in excellent agreement with the estimate of 0.83 derived from chlorophyll fluorescence yield

measurements (7). Previous work suggested that the reduced light harvesting efficiency of seg-

regated membranes was due to an inability of excitation initiated in LHCII aggregates to drive

photochemistry (3), resulting in the “disconnection” of LHCIIs from reaction centers. To ex-

plore this hypothesis we initiated excitation on each LHCII in both the mixed and segregated

membranes and calculated the resulting photochemical yield, ΦLHCII (Fig. 3A). Both mem-

branes have a wide distribution of ΦLHCII (Fig. 3B). 〈ΦLHCII〉 for the segregated membrane,

0.49, is significantly lower than that for the mixed membrane, 0.75. This difference can be

explained most simply by the relative diffusion lengths in each membrane environment. In the

9

Figure 3: Influence of grana membrane morphology on photochemical yield. (A) Excitationwas initiated at each LHCII in both the mixed (left) and segregated (right) morphologies. Thecolor of the LHCII indicates the fraction of excitation that results in productive photochemistry(ΦLHCII, see colorbar on far right) as simulated with our model. (B) Histograms representingthe distribution of ΦLHCII for the mixed and segregated membranes using the coloration from(A).

mixed membrane case, nearly all LHCIIs exist within an LD (25 nm) of a PSII-S. In the seg-

regated membrane, however, the LHCII aggregate(s) have a diameter comparable to LD (50

nm), resulting in substantial loss of excitation prior to reaching a PSII-S. Clearly, LHCIIs in

the segregated membrane show reduced light harvesting function; however, the LHCIIs do not

completely disconnect (ΦLHCII ≈ 0). To disconnect an LHCII aggregate from the neighboring

PSII-S requires the aggregate diameter to exceed 100 nm, which is unlikely to occur given that

a grana membrane disc has a diameter of ∼400 nm (2). The physiological benefit of segre-

gation remains unclear and may be explained by other aspects of photosynthesis, as has been

proposed (33, 34).

Our model accurately describes the energy transfer network of the grana membrane and

10

thus paves the way for understanding PSII light harvesting in all environmental conditions. In

intense sunlight, the rate of light absorption by the antenna exceeds the rate of trapping by

reaction centers, and the need for photoprotection arises. The mechanisms underlying the non-

photochemical quenching of excess excitation in the antenna remain controversial (8). Most

of the hypothesized mechanisms of quenching are based, by necessity, on measurements per-

formed on isolated pigment-protein complexes (11). However, it has been unclear whether the

mechanisms observed in vitro occur in vivo. Our model provides a unified framework for un-

derstanding to what extent picosecond dynamics observed in vitro explain in vivo chlorophyll

fluorescence data. More broadly, PSII is connected to the rest of photosynthesis through the

electron transport chain and the pH gradient across the thylakoid membrane (35,36). Our model

could be extended to include the unappressed portion of thylakoid membranes, which contains

photosystem I, and integrated into models of the entire thylakoid to fully model photosynthesis

in plants.

Methods

Overview

There has been an extensive theoretical development of energy transfer models in photosyn-

thetic complexes composed of a few proteins. Excitation dynamics can be calculated directly

from the Hamiltonian of a pigment-protein complex (37). However, exact calculations of the dy-

namics using the Hamiltonian are computationally costly, and, while feasible on complexes with

∼100 pigments, are impractical for simulating dynamics in the photosystem II-containing por-

tions of the thylakoid membrane, which contain hundreds of complexes and >10,000 pigments.

Another common technique for calculating excitation dynamics is by perturbative treatments of

the Hamiltonian, such as the Generalized Forster and Redfield methods (21). In photosystem II

(PSII), the energetic coupling between pigments spans a wide range, such that neither method is

11

appropriate on its own. Modified Redfield theory can interpolate between the strong and weak

coupling regimes, but it does not account for dynamic localization in which coupling between

the phonon modes of the protein and the pigments’ excited states restrict the delocalization of

excitonic states. Therefore, energy transfer in PSII has been treated using a combination of the

Modified Redfield and Generalized Forster theories (23,38,39). Transfer within tightly coupled

clusters of pigments is treated by Modified Redfield theory and transfer between these clusters

is treated using Generalized Forster theory.

In previous work on PSII supercomplexes (PSII-S), we used the Modified Redfield/Generalized

Forster (MR/GF) method in combination with a novel approach for defining highly coupled

clusters of chlorophylls, or domains, that optimizes the separation of timescales between intra-

and inter-domain transport (23). Using these domain definitions we were able to coarse-grain

the excitation dynamics at the domain level. Recent work using a more exact method (ZOFE)

for calculating energy transfer dynamics has further validated our MR/GF simulations (40).

Another group has performed HEOM calculations on a quadrant of the PSII-S (41). A compar-

ison of our MR/GF simulations with these calculations gives good agreement when appropriate

domain definitions are used.

We have used the same domain definitions in our simulations of energy transfer in thy-

lakoid membranes as were used for modeling PSII-S. We assumed that the timescales of en-

ergy transfer between domains on different complexes in the membrane are slow relative to

the intra-domain timescales (<1 ps−1) calculated previously (23) and used Generalized Forster

theory to calculate the rates between domains on different complexes (see Excitation energy

transfer section below). The locations of all of the PSII-S and major light harvesting complex II

(LHCII) antenna in a grana membrane were calculated on the basis of Monte Carlo simulations

of thylakoid membranes (24) (see Monte Carlo simulations of paired grana membranes section

below). The crystal structures of PSII-S (20) and LHCII (27) were overlaid on top of these

12

simulations (see Chlorophyll configurations section below).

The dynamics of PSII light harvesting can be represented by a kinetic network composed of

1st-order rate constants. The master equation formalism can be used to calculate the dynamics

of excitation:

P (t) = KP (t), (3)

where K is a rate matrix containing the first order rate constants of excitation transfer between

all compartments (which include all domains) in the network, and P (t) is the vector of compart-

ment populations (11, 42). Within this framework, we use simple models for charge separation

and trapping when excitation reaches the reaction center (see Electron transfer model section

below). While there are more complex models for electron transfer within the reaction cen-

ter (43), the two-compartment model used here was parameterized on PSII-S with different

antenna sizes (23) and is sufficient to determine the overall timescales of this process. The

non-radiative rate constant of decay from each domain was (2 ns)−1 (23). The fluorescence rate

constant for each exciton was scaled by its transition dipole moment squared with the average

fluorescence rate constant across all excitons set to (16 ns)−1 (23).

Solving eq. 1 for P (t), which contains the dynamics of excitation population on all com-

partments in the network, gives

P (t) = CetΛC−1P (0), (4)

where C is a matrix which contains the eigenvectors of K, Λ is a diagonal matrix containing

the eigenvalues of K, and P (0) is the initial vector of populations. Using eq. 2, we calculate

chlorophyll fluorescence decays, the dynamics of excitation over time, and the yields of the

different dissipation processes available to excitation - non-radiative decay, fluorescence, and

productive photochemistry in the reaction centers (see Calculations of P (t) section below).

13

Monte Carlo simulations of paired grana membranes

Grana-scale pigment-protein complex configurations were generated via computer simulations

of an extension of the model presented in ref. (24). Briefly, disc-shaped particles L representing

LHCII complexes and rod-shaped particles P representing so-called C2S2 PSII-S in 2d layers α

and β interacted via hard-core repulsive interactions, plus the attractive energetic potentials

ustack(rLα−Lβ) =

−εstack

(rLα−Lβ

−σL

σL

)2

if rLα−Lβ < σL

0 otherwise,(5)

uM(rLα−Pα) =

{−εM if rLα−Pα < λMσL

0 otherwise, and(6)

uagg(rLα−Lα) =

{−εagg if rLα−Lα < λaggσL

0 otherwise,(7)

with εstack = 4 kBT , εM = 2 kBT , λagg = 1.15, and all other parameters as in ref. (24).

The potentials in Eqs. 5 and 6 are motivated in ref. (24). The square-well attraction in

Equation 7 acts a phenomenological, non-specific energetic driving force for LHCII aggregation

(44). In the “mixed” condition, εagg = 0 kBT , while in the “segregated” condition, εagg = 1

kBT .

Canonical ensemble Metropolis Monte Carlo simulations of this model were performed as

in ref. (24). A ratio of free LHCII to PSII-S particles of 6 was chosen to match the conditions

of ref. (3), and a particle packing fraction of 0.75 was chosen to match typical grana conditions.

Thus, 64 PSII-S particles and 384 free LHCII particles were initialized in each of two square

boxes of side length 200 nm. Simulations were equilibrated with periodic boundary conditions

for at least 15M Monte Carlo sweep steps.

Chlorophyll configurations

Representative configurations from the stroma-side-up layers of the Monte Carlo simulations

were selected for analysis. For each configuration, chlorophyll coordinates were assigned for

14

each pigment-protein particle by aligning the center and axis of rotation of the chlorophyll coor-

dinates from refs. (20,27) to the center and axis of rotation of the simulated particle (Fig. S3). As

discussed and motivated in ref. (23), we have substituted the structure of an LHCII monomer in

the place of the minor light harvesting complexes in PSII-S. Because simulated LHCII particles

were radially symmetric, an axis of rotation was randomly selected for each LHCII particle.

In a small number of cases across the membrane, pigments belonging to one complex enter

into the excluded area of a different complex. This overlap originates from a mismatch between

the idealized geometries used for the Monte Carlo simulations and the real pigment structure

as shown in Fig. S2A. As a result of protein overlap, there exist a small number of analogously

high transfer rates within the membrane rate matrix. In order to understand the influence of such

rates on the overall description of transport in the membrane, we have removed transfer rates

exceeding certain thresholds from the rate matrix and re-calculated the resulting fluorescence

decay. As can be seen in Fig. S2B, this does not alter the fluorescence decay dynamics.

Excitation Energy Transfer Model

The rate of energy transfer from a donor domain d to an acceptor domain a, kdoma←d (eq. 3), is

typically calculated for each inhomogeneous realization of a pigment-protein complex using

generalized Forster theory (23, 38, 39). kdoma←d is the Boltzman-weighted (eq. 4) sum of the rates

from the excitons in d (|M〉 ∈ d) to the excitons in a (|N〉 ∈ a). The rates between excitons

are calculated using the generalized Forster equation (eq. 5-6), where∫∞

0dtAM(t)F ∗N(t) is

the overlap integral, |VM,N |2 is the electronic coupling between the two excitons, and Uµ,M

is the coefficient of the M th exciton on the site µ. Calculation of some observable of the

system, such as the fluorescence lifetime, is usually done for each realization and then averaged.

However, the thylakoid membrane contains >10,000 chlorophylls. Generating hundreds of

realizations of the population coefficients and the overlap integrals for the membrane is not

15

only computationally intensive, but may not necessary to accurately describe the dynamics of

the system at the protein length scale.

kdoma←d =

∑|M〉∈d|N〉∈a

kN←MP(d)M (8)

P(d)M =

e−EM/kBT∑|M〉∈d e

−EM/kBT(9)

kM←N =|VM,N |2

~2π

∫ ∞0

dtAM(t)F ∗N(t) (10)

|VM,N |2 = |∑µ,γ

Uµ,MHelµ,γUγ,N |2 (11)

Using the inhomogeneously averaged rate matrix gives the correct overall dynamics and is

representative of the energetics of PSII, while greatly increasing the computational efficiency.

It is computationally feasible to calculate a single rate matrix for the intact thylakoid mem-

brane using MR/GF theory through the use of supercomputers and patience. This calculation,

however, is likely to be unnecessary, because a much more computationally efficient method

is available for calculating the inhomogeneously averaged rate matrix, which gives the correct

overall dynamics, as shown by the fluorescence lifetime comparison in Fig. S1 for a PSII-S.

While the dynamics appear to be accurate, this averaging flattens out the energy landscape and

will speed up diffusion, as has been shown in quantum dot arrays (29). The standard devi-

ation of the exciton energies across the ensemble of inhomogeneous realizations in PSII are

significantly less than kBT , suggesting that excitation movement through the grana membrane

should not be greatly affected by averaging. Nonetheless, ongoing developments in making

exact calculations of energy transfer more scalable (45) will hopefully enable such calculations

on membranes in the future.

16

Directly calculating the inhomogeneous average transfer rates between domains substan-

tially reduces the computational burden because a single PSII-S contains all of the different

domains that occur throughout the grana membrane. The most computationally demanding

components for determining the rate matrix for a given inhomogeneous realization are the over-

lap integral in eq. 5 and the matrix to transform from the site to the exciton basis in eq. 6. We

have assumed that domain definitions, overlap integrals, and the transformation matrices for

PSII-S and LHCII are the same at the membrane level as they are in isolated complexes. As

a result these terms have been computed previously for several hundred realizations of inho-

mogeneous broadening for the four different types of transfers (from LHCII to LHCII, PSII to

PSII, LHCII to PSII and PSII to LHCII) in the membrane (23). We tabulated these values and

combined them with a calculation of the electrostatic coupling between sites (the only term that

must be calculated for each domain-to-domain rate in the membrane). By using our tabulated

values for the population matrices and overlap integrals, we have reduced the computational

time by 3-4 orders of magnitude, turning an otherwise burdensome calculation into one that can

be performed in less than 1 day on a single CPU. We note that this simplification works only

because we calculate the inhomogeneously averaged rate between each domain. This allows for

a much smaller sampling of the possible combinations of site energies then would be required

for even a single inhomogeneous realization of the entire thylakoid membrane.

Our algorithm for calculating 〈kdoma←d〉inhom. real. for each domain pair in the membrane was

as follows. We first checked if the two domains were within 60 A of each other. If yes, we

proceeded to calculate the average rate; if no, the rate was set equal to 0. We used 60 A as a

cutoff based on the distance dependence of rates in the PSII supercomplex. If the two domains

were from the same complex, we used the inhomogeneously averaged rate of transfer from

already performed generalized Forster/modified Redfield calculations (23). If not, we noted

in which type of complex the donor domain and the acceptor domain were in, either LHCII

17

or PSII-S. We calculated kdoma←d using eq. 3-6 with pre-tabulated values of the overlap integral,

the population matrices, and the energies of the excitons for a few hundred inhomogeneous

realizations. The electronic coupling Helµ,γ was uniquely calculated for that pair of domains

using the ideal dipole approximation. We then averaged over these tabulated inhomogeneous

realizations to get 〈kdoma←d〉inhom. real..

Electron transfer model

Both the identity of the primary donor and the kinetics and mechanism of charge separation in

the PSII reaction center remain controversial (46). The lack of agreement between the various

experimental results and theory has resulted in a variety of phenomenological and conceptual

models being used to interpret experimental data (17, 47). In previous work on PSII supercom-

plexes, we used the simplest kinetic model that describes the processes known to occur in the

PSII reaction center (23),

. (12)

Here, RC is the reaction center domain composed of the 6 pigments of the reaction center. The

“radical pair” states RP1 and RP2 and the rate constants kRC , kCS , and kirr are used to model

the electron transfer steps in the reaction center. RP1 and RP2 are non-emissive states that do

not have a direct physical analog with charge-separated states in the reaction center. Rather,

this approach allows us to describe the overall process of a reversible charge separation step

followed by an irreversible step and establish the approximate timescales of these events relative

to energy transfer in the light harvesting antenna. The rates were previously parameterized

using fluorescence decays of PSII supercomplexes of different sizes (19), with τCS = 0.64 ps,

18

τRC = 160 ps, and τirr = 520 ps (k = 1/τ ) (23).

Calculations of P (t)

Using the definitions established in the previous sections, we can write K as follows:

Kji =

kdomj←i , i, j ≤ Ndom, i 6= j (13)

kcs, i ∈ RC, j ∈ RP1, j > Ndom (14)

krc, i ∈ RP1, j ∈ RC, i > Ndom (15)

kirr, i ∈ RP1, j ∈ RP2, i, j > Ndom (16)

kdump, i ≤ Ndom, j = Ncomp − 1 (17)

kfli , i ≤ Ndom, j = Ncomp (18)

−∑k 6=i

kdomk←i, i = j, (19)

where Kij is a matrix element of row i and column j, Ndom is the total number of domains, and

Ncomp is the total number of compartments (Ndom plus all RP1, RP2, Fl, and dump compart-

ments). The sizes of the K for the mixed and segregated membranes modeled (Fig. 1A-B, main

text) were both 10882 x 10882, which meant that calculating the eigenvalues and eigenvectors

required for simulating the dynamics of excitation (eq. 2) required the use of supercomputers

with ∼30 GB of memory.

Fluorescence decays were calculated using the equations described in ref. (23). P (0) in all

cases was that for ChlA excitation. The photochemical yield was calculated by summing over

the populations in all RP2 states at t = 1 sec.

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23